Conventionally, a control apparatus has been known that executes control for automatically stopping, and then, automatically starting, an internal combustion engine (see, for example, Patent Document 1). While a vehicle is traveling, such a control apparatus cuts off fuel supply when the vehicle speed is greater than or equal to a predetermined vehicle speed within a predetermined vehicle speed range, to have the internal combustion engine stop automatically, and then, to have the internal combustion engine start automatically when the vehicle speed is below the predetermined vehicle speed. This configuration can make fuel efficiency performance improved because fuel supply to the internal combustion engine is cut off by automatically stopping the internal combustion engine while the vehicle is traveling.